Mathematica Moravica Vol. 18-2 (2014), 105–181 Transversal Functional Analysis Milan R. Tasković Short survey. This paper provides an introduction to the ideas and methods of transversal functional analysis based on the transversal sets theory. A unifying concept that lies at the heart of transversal functional analysis is that of a transversal normed linear space. I have developed the theory far enough to include facts of have called the three new basic principles of linear analysis as: Form of Hahn-Banach theorem, Form of Principle of Uniform Boundedness (= Form of Banach-Steinhaus theo- rem), and Form of Open Mapping theorem. In the classical functional analysis fundamental fact is Riesz lemma. In transversal functional analysis (on lower transversal normed spaces) its role play so-called Ge- ometrical lemma! This paper presents applications of the Axiom of Infinite Choice. 1. Transversal upper normed spaces Let X be a linear space over K (:= R or C). The mapping x x : X [a; b] for some 0 a < b < + or x x : X [a; b) for7! jj somejj 0 !a < b + is called≤ an upper1 transversal7! k k seminorm! (or upper ≤ ≤ 1 seminorm) iff: x a for every x X, λx = λ x for all λ K and x X, and if therejj jj ≥ is a function g :[2a; b)2jj [jja; b)jsuchj jj jj that 2 2 ! (Nu) x + y max x ; y ; g x ; y jj jj ≤ jj jj jj jj jj jj jj jj for all x; y X. n o Further,2x x is called an upper transversal norm (or upper norm) iff in addition:7! jjx jj= a if and only if x = 0. jj jj An upper transversal normed space (X; ) over K consists of a jj · jj linear space X over K together with an upper transversal norm x x . 7! jj jj 2010 Mathematics Subject Classification. Primary: 04A25, 05A15, 47H10, 54E15; Secondary: 54H25, 54E24. Key words and phrases. Transversal (upper, lower and middle) normed spaces, Transversal seminorms, Form of Hahn-Banach theorem, Form of Principle of Uniform Boundedness, Form of Banach-Steinhaus theorem, Form of Open Mapping theorem, Form of Riesz lemma, Geometrical lemma, Form of convergence principle, Lower compactness, Lower total continuous operators, Lower locally compactness, Lower Fredholm alternative, Extension Leray-Schauder principle, Peano’s equation, Lower compact operators. c 2014 Mathematica Moravica 105 106 Transversal Functional Analysis The function g :[a; b)2 [a; b) in (Nu) is called upper bisection fun- ction. From (Nu) it follows,! by induction, that there is a function M : [a; b)n [a; b) such that ! x0 xn jj − jj ≤ max x0 x1 ;:::; xn−1 xn ; M x0 x1 ;:::; xn−1 xn ≤ jj − jj jj − jj jj − jj jj − jj n o for all x0; x1; : : : ; xn X and for any fixed integer n 1. (The proof of this fact it follows immediately2 from induction!) ≥ It is easy to verify that every upper transversal normed linear space X is a transversal upper space (see: Tasković [2005]) with respect to the upper 0 transverse ρ : X X R+ defined by × ! ρ[x; y] = x y for all x; y X; jj − jj 2 thus we obtain ρ[x z; y z] = ρ[x; y] and ρ[λx, λy] = λ ρ[x; y] for all − − j j x; y; z X and for every scalar λ K. In this2 sense, an upper transversal2 normed space X is said to be upper complete if it is upper complete as a transversal upper space. The upper convergence xn x (n ) means xn x a (n ). We will in further! denote! 1 by G([a;jj b]) the− setjj ! of all upper! 1 bisection fun- ctions g :[a; b]2 [a; b] which are increasing satisfying g(t; t) t for every t [a; b]. ! ≤ 2We notice that upper transversal norm x x is a general convex fun- ction. The proof is simple. See this chapter7! of the jj jj book. Example 1 (Upper intervally normed space). Let X be a linear space over K and let G be the set of all nonincreasing, left continuous functions x Mx : R [a; b] 0 7! ! for a < b (a; b R+), where inf Mx = a and sup Mx = b. Then (X; G) is an upper 2 transversal normed space if: Mx(0) = a for every x X, Mx = b for x 0 and 2 ≤ Mx = a for x > 0, σ Mλx(σ) = Mx for every x X λ 2 j j and for all σ R and λ = 0, and if there is a function g :[a; b]2 [a; b] for 0 2 6 ! a; b R+ (a < b) such that the following ineuality holds in the form as 2 Mx+y(p + q) max Mx(p);My(q); g Mx(p);My(q) ≤ for all x; y X and for all p; q 0n. o 2 ≥ The fundamental results of this section are the statements characterizing finite dimensional spaces (as a Riesz’s theorem type), statement of separable upper transversal normed spaces and a statement as the form of Riesz lemma for upper transversal normed spaces. See brief proofs for this in: Tasković [2005]. Also, in this part, we derive a general extension theorem, as a form of Hahn-Banach theorem, for linear functionals on an arbitrary linear space. Milan R. Tasković 107 In the next we can apply this theorem to the problem of the existence of upper bounded linear functionals. If in the preceding definition of transversal upper normed spaces the con- dition of homogenity of the form: λx = λ x for all λ K and x X is k k j j k k 2 2 to exchange with the condition of the form: λx = f(λ) x for all λ K, k k j jk k 2 x X, and a function f : K K, then we have a transversal upper global2 normed space (X; !), where x x is a transversal upper global norm. Adequate we obtaink · k a transversal7! k upperk global semi-normed space as and transversal upper global seminorm. On the other hand, in an upper transversal normed space X with the upper bisection function g G([a; b)), the following inequality holds as 2 n n (1) xk xk ≤ k k k=1 k=1 X X for all points x1; : : : ; xn X , or in a precisely form as the following inequality which is very practical in2 further as n (1’) xk max x1 ;:::; xn ≤ k k k k k=1 n o X for all points x1; : : : ; xn X . (The proof of inequalities (1) and (1’) via induction may be found in:2 T a s k o v i ć [2005].) Otherwise, a fundamental first example of transversal upper normed spaces 0 2 0 with the upper bisection function g :(R+) R+ defined by g(s; t) := s+t, ! is a classical normed space.1 Example 2 (The space Lp(a; b) for p R 0 ). If in the set Lp(a; b) for p R 0 introduced, in the classical style, the following2 nf g compositions in the form as2 nf g (x + y)(t) = x(t) + y(t); and (λx)(t) = λx(t); then we obtain that Lp(a; b) is a vector space. The zero vector for this space is a function which is almost everywhere equel to zero. This space is a transversal 1History of normed spaces. J e a n D i e u d o n n é: “Starting from the work of F r i e d r i c h R i e s z and E d u a r d H e l l y on the problem of moments inertia it was a natural generalization to define norms on arbitrary vector spaces. This was done independently by S t e f a n B a n a c h and H a n s H a h n about 1920. In 1932 Banach published a book containing a comprehensive account of all results known at that time in the theory of normed spaces, and in particular the theorems he had published in his papers of 1922 and 1929. A large part was devoted to the concept of weak convergence and its generalizations, which he had begun to study in 1929. This book had on Functional Analysis the same impact that Van der Waerden’s book had on Algebra two years earlier. Analysts all over the world began to realize the power of the new methods and to apply them to a great variety of problems; Banach’s terminology and notations were universally adopted, complete normed spaces became known as Banach spaces, and soon their theory was considered as a compulsory part in most curricula of graduate students. After 1935, the theory of normed spaces became part of the more general theory of locally convex spaces”. 108 Transversal Functional Analysis upper normed space if the transversal upper norm x x is defined with 7! k k 1=p b (2) x = x(t) pdt k k j j Za ! for every x Lp(a; b) and for an arbitrary fixed p R 0 . Then to arise that (2) has all attributes2 of transversal upper norm, where2 inequalitynf g (Nu) is the form as 1=p b p x + y max x ; y ; x(t) + y(t) dt k k ≤ 8k k k k a j j j j ! 9 < Z = for all x; y Lp(a; b) and p :R 0 . (We notice, in the special case; for p 1, the inequality (Nu)2 may be in something2 nf g differently form.) ≥ Something dilatation.